Lecture 22. Ideal Bose and Fermi gas (Ch. 7) Gibbs factor the grand partition function of ideal quantum gas: fermions: ni = 0 or 1 bosons: ni = 0, 1, 2, ..... Outline Fermi-Dirac statistics (of fermions) Bose-Einstein statistics (of bosons) Maxwell-Boltzmann statistics Comparison of FD, BE and MB.

The Partition Function of an Ideal Fermi Gas The grand partition function for all particles in the ith single-particle state (the sum is taken over all possible values of ni) : If the particles are fermions, n can only be 0 or 1: Putting all the levels together, the full partition function is given by: The partition functions of different levels are multiplied because they are independent of one another (each level is an independent thermal system, it is filled by the reservoir independently of all other levels).

Fermi-Dirac Distribution The probability of a state to be occupied by a fermion: The mean number of fermions in a particular state: Fermi-Dirac distribution ( is determined by T and the particle density)

Fermi-Dirac Distribution At T = 0, all the states with  <  have the occupancy = 1, all the states with  >  have the occupancy = 0 (i.e., they are unoccupied). With increasing T, the step-like function is “smeared” over the energy range ~ kBT. 1 ~ kBT T =0  =  (with respect to ) The macrostate of such system is completely defined if we know the mean occupancy for all energy levels, which is often called the distribution function: While f(E) is often less than unity, it is not a probability: n=N/V – the average density of particles

The Partition Function of an Ideal Bose Gas The grand partition function for all particles in the ith single-particle state (the sum is taken over all possible values of ni) : If the particles are Bosons, n can be any #, i.e. 0, 1, 2, … The partition functions of different levels are multiplied because they are independent of one another (each level is an independent thermal system, it is filled by the reservoir independently of all other levels). Putting all the levels together, the full partition function is given by:

Bose-Einstein Distribution The probability of a state to be occupied by a Boson: The mean number of Bosons in a particular state: Bose-Einstein distribution The mean number of particles in a given state for the BEG can exceed unity, it diverges as   min().

Comparison of FD and BE Distributions Maxwell-Boltzmann distribution:

Maxwell-Boltzmann Distribution (ideal gas model) Recall the Boltzmann distribution (ch.6) derived from canonical ensemble: The mean number of particles in a particular state of N particles in volume V: Maxwell-Boltzmann distribution MB is the low density limit where the difference between FD and BE disappears.

Comparison of FD, BE and MB Distribution

Comparison of FD, BE and MB Distribution (at low density limit) The difference between FD, BE and MB gets smaller when  gets more negative. MB is the low density limit where the difference between FD and BE disappears.

Comparison between Distributions Bose Einstein Fermi Dirac Boltzmann indistinguishable Z=(Z1)N/N! nK<<1 spin doesn’t matter localized particles  don’t overlap gas molecules at low densities “unlimited” number of particles per state indistinguishable integer spin 0,1,2 … bosons wavefunctions overlap total  symmetric photons 4He atoms unlimited number of particles per state indistinguishable half-integer spin 1/2,3/2,5/2 … fermions wavefunctions overlap total  anti-symmetric free electrons in metals electrons in white dwarfs never more than 1 particle per state

“The Course Summary” Ensemble Macrostate Probability Thermodynamics micro-canonical U, V, N (T fluctuates) canonical T, V, N (U fluctuates) grand canonical T, V,  (N, U fluctuate) (the Landau free energy) is a generalization of F=-kBT lnZ The grand potential the appearance of μ as a variable, while computationally very convenient for the grand canonical ensemble, is not natural. Thermodynamic properties of systems are eventually measured with a given density of particles. However, in the grand canonical ensemble, quantities like pressure or N are given as functions of the “natural” variables T,V and μ. Thus, we need to use to eliminate μ in terms of T and n=N/V.